SIGMA 22 (2026), 054, 31 pages      arXiv:2011.01800      https://doi.org/10.3842/SIGMA.2026.054
Contribution to the Special Issue on Interactions of Poisson Geometry, Lie Theory and Symmetry in honor of Rui Loja Fernandes

The Geometry of Loop Spaces III: Isometry Groups of Contact Manifolds

Satoshi Egi ^a, Yoshiaki Maeda ^b and Steven Rosenberg ^c
a) Rakuten Institute of Technology, Rakuten, Inc., Japan
^b) Tohoku Forum for Creativity, Tohoku University, Japan
^c) Boston University, USA

Received April 25, 2025, in final form May 03, 2026; Published online May 29, 2026

Abstract
Let $M_p$ be a circle bundle with first Chern class $p[\omega]$ over a closed $4n$-dimensional integral symplectic manifold $\bigl(\overline{M},\omega\bigr)$. Equivalently, $M_p$ is a closed contact $(4n+1)$-manifold whose Reeb orbits are all closed and have the same period. For a metric $g$ on $M_p$ compatible with the symplectic structure and the geometry of the circle fiber, we use Wodzicki-Chern-Simons forms on the loop space $LM_p$ to prove that $\pi_1({\rm Isom}(M_p,g))$ is infinite for ${|p| \gg 0}$. We also give the first high-dimensional examples of nonvanishing Wodzicki-Pontryagin forms.

Key words: contact manifolds; Wodzicki-Chern-Simons classes; isometry groups.

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